Turbulence and CFD models:
Theory and applications
1
Roadmap to Lecture 4
1. Practical turbulence estimates
2
Practical turbulence estimates
Introduction
• In Lecture 3, Kolmogorov scales, Taylor scales, and integral scales were introduced.
• We then explored the concepts of energy spectrum, energy cascade, integral length
scale, and grid length scale.
• We also studied the basic concepts of turbulence near the wall, we introduced the
Law of the Wall, and the non-dimensional quantity y+.
• Finally, we took a glimpse to a turbulence model.
• At this point, the question is,
• How can we use this information?
• How can we get an initial estimate of the variables related to the turbulence
model?
• How can we estimate the meshing requirements?
• Hereafter, we will give some standard practices on how to get turbulence estimates.
• Have in mind that many commercial applications have these relationships already
implemented, so you do not need to do these computations (which are trivial).
3
Practical turbulence estimates
Introduction
• Using everything we have learned so far, we can get global estimates for the
following variables:
• Eddies velocity, eddies size, and time scales (integral, Taylor, and Kolmogorov).
• Rough estimates of the number of grid points or cells needed.
• Energy dissipation rate .
• Turbulent kinetic energy
.
• Turbulent kinematic viscosity
• Turbulent intensity
.
.
• Wall distance units y+.
• Remember, we will compute initial estimates for global quantities.
• If you want to get the local values, you will need to run a simulation where you will
resolve a transport equation for these variables.
• I cannot stress this enough; we will compute rough estimates which are fine for initial
conditions or generating an initial mesh.
4
Practical turbulence estimates
Introduction
• Let us first compute the integral eddy length scale, turbulence intensity, turbulent
kinetic energy, and turbulent dissipation.
• We will use the LIKE acronym [1] to describe the workflow that we will use to
compute these practical estimates.
• L
=
=
integral eddy length scale
• I
=
=
turbulence intensity
• K
=
=
turbulent kinetic energy
• E
=
=
turbulent dissipation
• We already know many relations from Lecture 3.
• We will introduce a few new equations.
• Many of these relationships can be derived from dimensional analysis.
• Remember to always check the dimensional groups.
[1] S. Rodriguez. “Applied Computational Fluid Dynamics and Turbulence modeling”. Springer, 2019.
5
Practical turbulence estimates
A reminder about the units
Derived quantity
Symbol
Dimensional units
SI units
Velocity
LT-1
m/s
Density
ML-3
kg/m3
Kinematic viscosity
L2T-1
m2/s
Dynamic viscosity
ML-1T-1
kg/m-s
Energy dissipation rate per unit mass
L2T-3
m2/s3
Turbulent kinetic energy per unit mass
L2T-2
m2/s2
Length scales
L
m
Wavelength
L-1
1/m
Intensity
-
-
Stress
ML-1T-2
kg/(m-s2) or N/m2 or Pa
6
Practical turbulence estimates
Integral eddy length scale – LIKE
•
Usually, the integral length scales
domain,
are represented by a characteristic dimension of the
•
That is, the system characteristic length places a limit on the maximum integral eddy length.
•
In practice, this limit is not reached nor there is a typical value.
•
Therefore, conservative approximations are often used based on a percentage of the system
characteristic length.
•
For example,
•
If you are simulating the flow about a cylinder, you can say that the largest eddies are
about 70% of the cylinder diameter.
•
If you are simulating the flow in a pipe, you can say that the largest eddies are about 90%
the diameter of the pipe.
•
If you are simulating the flow about an airfoil (with no large flow separation), you can say
that the largest eddies are about the airfoil thickness.
8
Practical turbulence estimates
Integral eddy length scale – LIKE
• You will find often the following relationships in the literature.
• For internal flows (pipes and ducts), where D is the diameter or height,
• For boundary layers over surfaces, where
thickness,
is the turbulent boundary layer
where the boundary layer thickness can be approximated using the
following correlation (among many available in the literature),
9
Practical turbulence estimates
Integral eddy length scale – LIKE
• You will find often the following relationships in the literature.
• For grid generated turbulence in wind tunnels, where S is the grid spacing,
• The following is a personal estimate that I often use,
•
•
•
Where hb is the blockage height in the direction of the incoming flow.
For example:
• Airfoil thickness, if it is aligned with the flow or at a low AOA.
• If the airfoil is at a high AOA, the blockage height.
• Frontal area of a body (cylinder, truck, and so on).
This relationship can be use for internal and external flows
10
Practical turbulence estimates
Integral eddy length scale – LIKE
• If you have estimates for
follows.
and
, you can compute the integral length scales as
• Taylor suggests [1] that the integral length scales can be approximated as
follows,
Use this estimate if you are interested in
the largest integral length scale
• The previous estimate has been be improved by using experimental data, as
explained by Wilcox [2],
where
Use this estimate if you are
interested in the average integral
length scale
This is the preferred correlation
[1] G. I. Taylor. Statistical theory of turbulence. Proceedings of the Royal Society of London. 1935.
[2] D. Wilcox. Turbulence Modeling for CFD. DCW Industries Inc., 2010.
11
Practical turbulence estimates
Turbulence intensity – LIKE
• The turbulence intensity (also called turbulence level) is often abbreviated as follows,
• The turbulence intensity can be computed as follows,
Intensity of velocity fluctuations
Mean velocity – Freestream velocity
• The intensity of the velocity fluctuations (turbulence strength) is defined by the root
mean square (RMS) of the velocity fluctuations,
•
•
It gives a measure of the dispersion of the
velocity fluctuations squared ( or normal
Reynolds stresses).
It is nothing else but the standard deviation
of the fluctuations.
12
Practical turbulence estimates
Turbulence intensity – LIKE
• The Reynolds stress components
,
,
, can also be regarded as the kinetic
energy per unit mass of the fluctuating velocity in the three spatial directions.
• If we sum the normal Reynolds stresses and multiply by 0.5, we obtain the turbulent
kinetic energy,
• The normal Reynolds stresses can be normalized relative to the mean flow velocity,
as follows,
These three quantities are
known as relative intensities.
13
Practical turbulence estimates
Turbulence intensity – LIKE
•
For anisotropic turbulence (i.e., the normal components of the Reynolds stress tensor are
unequal), a rough but useful estimate to the normal Reynolds stress components is the
following,
Correlation based on flat-plate boundary layer
•
And recall that,
Turbulence intensities for a flat-plate boundary layer of thickness
[1] P. Klebanoff. “Characteristics of Turbulence in a Boundary Layer with Zero Pressure Gradient”. NACA TN 1247, 1955.
[1].
14
Practical turbulence estimates
Turbulence intensity – LIKE
•
For the case of isotropic turbulence, the normal components of the Reynolds stress tensor are
the same.
•
Then, the turbulence intensity can be computed as follows,
•
And recall that,
For isotropic
turbulence
[1] P. Klebanoff. “Characteristics of Turbulence in a Boundary Layer with Zero Pressure Gradient”. NACA TN 1247, 1955.
15
Practical turbulence estimates
Turbulence intensity – LIKE
• To use the previous relations, you need to have some form of measurements or
previous experience to base the estimate on.
• The turbulence intensity can also be estimated using empirical correlations.
• For instance, if you are working with pipes, there are many correlations that are
expressed in the form of a power law,
Hydraulic Reynolds number
• One widely used correlation is the following one,
• Remember, there are many forms of these correlations (for smooth and rough pipes).
16
Practical turbulence estimates
Turbulence intensity – LIKE
• If you are working with external aerodynamics, it might be a little bit more difficult to
get rule of thumb estimates.
• However, the following estimates are acceptable,
Low
Medium
High
1.0 %
5.0 %
10.0 %
Low turbulence intensity (1%): external flow around cars, ships, submarines, and aircrafts. Very high-quality wind-tunnels can
also reach low turbulence levels, typically below 1.0%.
Medium turbulence intensity (5%): flows in not-so-complex devices like large pipes, fans, ventilation flows, wind tunnels, low
speed flows, and fully-developed internal flows. Typical values are between 2.0% and 7.0%.
High turbulence intensity (10%): high-speed flow inside complex geometries like heat-exchangers and rotating machinery
(turbines and compressors). Typical values are between 10.0% and 20.0%.
17
Practical turbulence estimates
Turbulent kinetic energy – LIKE
• The turbulent kinetic energy
is also known as TKE.
• If you have some estimates of the velocity fluctuations or normal Reynolds stresses ,
the TKE (per unit mass) can be computed as follows,
• Otherwise, you can use a rule of thumb turbulence intensity estimate and compute
TKE as follows,
where
• Instead of using
freestream value
is the freestream velocity
, you can also use a value slightly higher that we will call turbulent
,
Arbitrary constant to correct velocity fluctuations
18
Practical turbulence estimates
Energy dissipation rate – LIKE
• Once TKE is known, together with a crude estimate of the integral eddy length scale,
the energy dissipation rate (per unit mass) can be computed as follows,
where
• You can compute the specific dissipation rate once you know
and , as follows,
where
• You can compute the integral eddy length scale from specific dissipation rate as
follows,
20
Practical turbulence estimates
Turbulent viscosity
• So far, we computed the integral eddy length scale, turbulence intensity, turbulent
kinetic energy and energy dissipation rate.
• In the previous lecture, we saw that in turbulence modeling there is an extra
ingredient, turbulent viscosity .
• We can also get an estimate for this quantity. How do we estimate it depends on the
turbulent model.
• For example, the
model computes the turbulent viscosity as follows,
Where you can compute
from
as follows,
• As usual, check the dimensional groups.
• Remember, the turbulent viscosity is not a physical quantity.
• Also, the turbulent viscosity is larger than the laminar viscosity (molecular viscosity).
21
Practical turbulence estimates
Turbulent viscosity
• We can also use a rule of thumb to get a fast estimate of the turbulent viscosity.
• Using the turbulence intensity
as follows,
, we can get an estimate for the viscosity ratio
Low
Medium
High
1.0 %
5.0 %
10.0 %
1-2
10
100
Low turbulence intensity (1%): external flow around cars, ships, submarines, and aircrafts. Very high-quality wind-tunnels can
also reach low turbulence levels, typically below 1.0%.
Medium turbulence intensity (5%): flows in not-so-complex devices like large pipes, fans, ventilation flows, wind tunnels, low
speed flows, and fully-developed internal flows. Typical values are between 2.0% and 7.0%.
High turbulence intensity (10%): high-speed flow inside complex geometries like heat-exchangers and rotating machinery
(turbines and compressors). Typical values are between 10.0% and 20.0%.
• We usually use these estimates when dealing with external aerodynamics.
22
Practical turbulence estimates
Estimation of y+ and wall distance units
23
Practical turbulence estimates
Estimation of y+
•
•
•
Let us revisit the definition of y+.
y+ is the non-dimensional distance normal to the wall.
It is very similar to the Reynolds number, but it is non-dimensionalized using the
shear velocity U .
wall shear stresses
In the previous equations:
•
•
•
y is the distance normal to the wall.
U is the shear velocity,
relates the mean velocity to the shear velocity.
24
Practical turbulence estimates
Estimation of y+
• y+ is very important quantity in wall bounded turbulence modeling.
• We can use y+ to estimate the mesh resolution near the wall before running the
simulation.
• Have in mind that we do not know a-priori the wall shear stresses at the walls.
• Therefore, we need to use correlations to get a rough estimate and generate the
initial mesh.
• The initial mesh is then generated according to the chosen near the wall
treatment, namely, wall resolving, wall functions, or y+ insensitive.
• Then, we run a precursor simulation to validate the mesh or to get a better
estimate of y+ and determine if we need a finer or coarser mesh.
• It is an iterative process, and it can be very time consuming, as it might require
remeshing and rerunning the simulation to satisfy the near the wall treatment.
25
Practical turbulence estimates
Estimation of y+
•
It is worth mentioning that y+ can be defined in
reference to the cell center or the cell nodes.
•
From now on, we are going to assume that we
are using a cell-centered solver.
•
•
Ansys Fluent, StarCCM+, OpenFOAM,
code Saturne, SU2, are all cell-centered.
•
It is up to you to check with the solver’s
documentation to find out with what kind of
formulation are you using.
By the way, y+ is computed in the same way
disregarding of the cell type used.
•
In the cell-centered formulation you need
to know the location of the cell center.
•
In the node-centered formulation you need
to know the location of the reference node.
Usually, the node closest to the surface.
26
Practical turbulence estimates
Estimation of y+
• At meshing time, to estimate the normal distance from the wall to the first cell center
(y), we use the well-known y+ definition.
• Where we set a target y+ value and then we solve for the quantity y.
• If you choose a low target y+ (e.g., less than 10), you will have a mesh that is
clustered towards the wall (small value of y).
• If you choose a large y+ value (e.g., more than 100), you will have a coarse
mesh close to the walls (large value of y).
WALL
WALL
Fine mesh towards the walls
Coarse mesh towards the walls
Note: depending on the implementation you can talk about cell center or node center.
27
Practical turbulence estimates
Estimation of y+
• At meshing time, to estimate the normal distance from the wall to the first cell center,
we use the well-known y+ definition,
• The problem is that at meshing time we do not know the value of the shear velocity,
• So, how do we get an initial estimate of this quantity?
• We can use correlations, as described in the next slide.
28
Practical turbulence estimates
Estimation of y+
•
At meshing time, to estimate the normal distance from the wall to the first cell center,
you can proceed as follows,
1.
2.
Compute the Reynolds number using the characteristic length of the
problem.
Compute the friction coefficient using any of the correlations available in
the literature. There are many correlations available ranging from pipes to
flat plates, for smooth and rough surfaces.
This correlation corresponds to a smooth flat plate case, ideal for external
aerodynamics.
3.
Compute the wall shear stresses using the friction coefficient computed in
the previous step.
4.
Compute the shear velocity using the wall shear stresses computed in the
previous step.
5.
Set a target y+ value and solve for y using the flow properties and previous
estimates.
This value represents the distance from the wall to the first cell center.
At this point, you can generate your mesh trying to respect this value.
•
You will find a simple calculator for the wall distance estimation in the following link: http://www.wolfdynamics.com/tools.html?id=2
29
Practical turbulence estimates
y+ additional comments – When to use wall functions
•
Most of the times you will find in the literature a y+ upper limit
of 200-300 for the applicability of wall functions.
•
In reality, this upper limit depends on the Reynolds number,
as shown in the figures.
•
For high Reynolds number you can reach y+ values of up to
1000.
•
Care must be taken if the y+ upper limit is below or close to
100.
•
•
Dimensionless mean velocity profile u+ as a function of the
dimensionless wall distance y+ for turbulent pipe flow with
Reynolds numbers between 4000 and 3600000 [1].
In this case, you should not use wall functions
because the overlap region is too narrow and is likely
that you will cluster the cell centers in the buffer region
or too close to it.
Remember, when using wall functions it is recommended to
cluster at least 5 cells in the log-law region in order to
resolve the velocity and TKE profiles.
Mean velocity profiles in pipe flow showing the collective approach
to a log law. The curves are for Reynolds numbers between Re =
31 x 103 and Re = 18 x 106 [2].
[1] F. Nieuwstadt, B. Boersma, J. Westerweel. Turbulence. Introduction to Theory and Applications of Turbulent Flows. Springer. 2016.
[2] B. McKeon, J. Li, W. Jiang, J. Morrison, A. Smits. Further observations on the mean velocity distribution in fully developed pipe flow. 2004.
30
Practical turbulence estimates
y+ guidelines – Correction of the mesh
• Remember, the thickness of the thermal sublayer for a high Prandtl number fluid
(e.g., water) is much less than the momentum sublayer thickness.
• Therefore, the mesh requirements close to the walls need to be corrected for the
thinner thermal boundary layer.
• The y+ value can be corrected as follows,
• A similar situation exists when working with multispecies transport and using the
Schmidt number (Sc).
• If the Schmidt number is considerably larger than unity, then the thickness of the
thermal diffusion sublayer is much less than the momentum sublayer thickness.
• In this situation, the y+ value can be corrected as follows,
31
Practical turbulence estimates
y+ guidelines – Limiting values and solution monitoring
• y+ always needs to be monitored during the simulation.
• Have in mind that it is quite difficult (if not impossible) to get a uniform y+
value at the walls.
• We usually monitor the average y+ value. If this value covers approximately
80% of the wall, we can take the mesh as a good one.
• Otherwise, we need to refine or coarse the mesh to get a more uniform
distribution of y+.
• It is also important to monitor the maximum values of y+. It is not a got
practice to have values larger than 1000.
• Values of y+ up to 300 are fine.
• Values of y+ larger than 300 and up to a 1000 are acceptable if they do not
cover a large surface area (no more than 20% of the total wall area), or if
they are not located in critical zones.
• It is also important to monitor the minimum y+, as some models might have
problems with low y+ values.
• Use common sense when accessing y+ value.
32
Practical turbulence estimates
Wall distance units x+ – y+ – z+
• Similar to , the wall distance units can be
computed in the stream-wise (
) and
span-wise (
) directions.
• The wall distance units in the stream-wise
and span-wise directions can be computed
as follows:
• And recall that
center, therefore:
is computed at the cell
where
Viscous length
33
Practical turbulence estimates
Wall distance units x+ – y+ – z+
•
It is not only about the y+ value.
•
In SRS simulations x+ and z+ are also important.
•
The x+ and z+ values strongly depend on the surface mesh aspect ratio.
•
The aspect ratio is defined as the ratio between the longest edge and shortest edge.
•
Recommended values of surface mesh aspect ratio for SRS simulations are 10 or less.
•
Meshing applications usually have adequate tools for mesh diagnostics.
•
This is a rough estimate based on experience.
34
Practical turbulence estimates
Wall distance units x+ – y+ – z+
•
Besides the surface aspect ratio, you should also check the ratio between surface cell base
length and the cell height. This aspect ratio can be approximated as follows,
•
Recommended values are 20 or less.
•
As you can see, this aspect ratio can impose strong requirements on the mesh.
•
Again, this is a rough estimate based on experience, and that should be used when conducting
SRS simulations
35
Practical turbulence estimates
Wall distance units x+ – y+ – z+
•
The wall distance units x+ and z+ values strongly depend on the surface mesh aspect ratio.
Surface mesh with large aspect ratio and large ratio
between surface cell base length and the cell height.
Surface mesh with low aspect ratio and low-to-moderate
ratio between surface cell base length and the cell height
36
Practical turbulence estimates
Wall distance units x+ – y+ – z+
• The stream-wise and span-wise wall distance units requirements are important when
conducting scale-resolving simulations (DES, LES, DNS).
• Typical requirements for LES are (these are recommendations based on different
references):
for
Wall resolving
for
Wall modeling
• As you can see, these requirements translate in much finer surface meshes and low
surface aspect ratios.
• We will talk more about this in Lecture 10.
• RANS simulations only have requirements respect to y+. They do not have strict
requirements when it comes to the span-wise and stream-wise directions.
• Therefore, they are more affordable.
37
Practical turbulence estimates
Wall distance units x+ – y+ – z+
• These strict requirements in SRS simulation are due to the fact that we need to
resolve the structures close to the walls (streaks).
• And this require fine meshes in all space directions.
Incoming flow
Incoming flow
Wall bounded flow – Channel flow
http://www.wolfdynamics.com/training/turbulence/channel1.mp4
38
Practical turbulence estimates
Wall distance units – A few mesh resolution guidelines and rough estimates
• The mesh is everything in CFD, and when it comes to turbulence modeling, it is
extremely important to have meshes with good quality and acceptable resolution.
• Some general guidelines for meshes to be used with RANS/DES/LES:
• Resolve well the curvature.
• Allow a smooth transition between cells of different sizes (at least 3 cells).
• Identify the integral scales and try to cluster at least 5 cells in the domain
regions where you expect to find these scales.
• If you have no restrictions, use a wall resolving approach, that is, fine
meshes in the direction normal to the walls.
• It is recommended to avoid hanging nodes (in Lecture 3 we illustrated what
they are).
39
Practical turbulence estimates
Wall distance units – A few mesh resolution guidelines and rough estimates
• The mesh is everything in CFD, and when it comes to turbulence modeling, it is
extremely important to have meshes with good quality and acceptable resolution.
• Some guidelines specific to RANS meshes:
• When it comes to RANS, the most important metric for mesh resolution is
the y+ value.
• Choose your wall treatment and mesh your domain according to this
requirement.
• If you are doing 3D simulations, there are no strict requirements when it
comes to the span-wise and stream-wise directions.
• But as a rule of thumb rule, you can use
and
values as high as
300 times the value of
and less than a 1000 wall distance units.
40
Practical turbulence estimates
Wall distance units – A few mesh resolution guidelines and rough estimates
• Some guidelines specific to LES meshes:
• When it comes to LES meshes, it is recommended to use wall functions.
• Otherwise, the meshing requirements are similar to those of DNS.
• Recommended wall distance units values are,
for
Wall resolving
for
Wall modeling
• If you are doing DNS simulations, the requirements for wall distance units in all
directions are in the order of 1.
• You might be able to go as high as 10 for
and
.
41
Practical turbulence estimates
Wall distance units – A few mesh resolution guidelines and rough estimates
• Some guidelines specific to DES meshes:
• Usually in DES simulations wall resolving meshes are used with RANS
requirements similar to RANS.
• That is, there are no specific requirements on the streamwise and spanwise
wall units values.
• However, it is extremely important to resolve well the integral length scales,
as in LES simulations.
• What is very tricky in DES, is how to control the transition from the RANS
mesh close to the walls to the fine-medium mesh far from the walls.
• You might encounter a mismatch between the RANS mesh and the LES
mesh, because the RANS mesh tends to be coarser in the streamwise and
spanwise directions.
• This transition, if not well controlled can cause grid induced separation and
stability problems.
42
Practical turbulence estimates
Profiles close to the walls – Dependence
on the y+ value
43
Practical turbulence estimates
Profiles close to the walls – Dependence on the y+ value
•
Distance normal to wall y (in meters).
•
Each circle represents a cell center.
44
Practical turbulence estimates
Profiles close to the walls – Dependence on the y+ value
•
Velocity profile close to the wall – y vs. U
•
Each circle represents a cell center.
45
Practical turbulence estimates
Profiles close to the walls – Dependence on the y+ value
•
Velocity profile close to the wall – y vs. U
•
Each circle represents a cell center.
46
Practical turbulence estimates
Profiles close to the walls – Dependence on the y+ value
•
Non-dimensional distance normal to wall y+.
•
Each circle represents a cell center.
47
Practical turbulence estimates
Profiles close to the walls – Dependence on the y+ value
•
Velocity profile close to the wall – y+ vs. U
•
Each circle represents a cell center.
48
Practical turbulence estimates
Profiles close to the walls – Dependence on the y+ value
•
Velocity profile close to the wall – y+ vs. U
•
Each circle represents a cell center.
49
Practical turbulence estimates
Profiles close to the walls – Dependence on the y+ value
•
Normalized velocity profile close to the wall – U+ vs. y+
•
Each circle represents a cell center.
50
Practical turbulence estimates
Profiles close to the walls – Dependence on the y+ value
•
Turbulent production and dissipation close to the walls.
•
Each circle represents a cell center.
51
Practical turbulence estimates
Profiles close to the walls – Dependence on the y+ value
•
Turbulent production and dissipation close to the walls.
•
Each circle represents a cell center.
52
Practical turbulence estimates
Summary of turbulence length scales
53
Practical turbulence estimates
Summary of turbulence length scales
• The Kolmogorov scales are summarized as follows,
Length scale
Time scale
Velocity scale
• The Taylor microscales are summarized as follows,
Length scale
Time scale
Velocity scale
54
Practical turbulence estimates
Summary of turbulence length scales
• Taylor [1] suggests that the integral length scales can be approximated as follows,
• This estimate can be improved by using experimental data [2],
where
• You can express the previous relation in function of
where
[1] G. I. Taylor. Statistical theory of turbulence. Proceedings of the Royal Society of London. 1935.
[2] D. Wilcox. Turbulence Modeling for CFD. DCW Industries Inc., 2010.
as follows,
and
55
Practical turbulence estimates
Summary of turbulence length scales
• The eddies turnover time is the ratio between the integral length scales and the
velocity
(the measure of the velocity fluctuations around the mean), and it can
be computed as follows,
• The eddy turnover time is a measure of the time it takes an eddy to interact with its
surroundings.
• The integral eddy velocity is the ratio of its integral length scale and its turnover time,
• If you assume isotropic turbulence then,
56
Practical turbulence estimates
Summary of turbulence length scales
• The different length scales can be related as follows,
•
Support equations:
57
Practical turbulence estimates
Summary of Reynolds numbers
• The different Reynolds numbers, based on different length scales, can be
summarized as follows,
Flow Reynolds number
Taylor Reynolds number
Kolmogorov Reynolds number
• The turbulent Reynolds number is related to the integral scales. It is a few order of
magnitude lower that the flow Reynolds number (on the order of 100 to 10000).
• The turbulence and Taylor Reynolds numbers can be related as follows
58
Practical turbulence estimates
Summary of turbulence length scales
•
During the previous lectures, sometimes
we used the notation
and sometimes
we used the notation .
•
represents the largest integral
eddy.
•
represents the average of the
integral eddies.
•
Some authors use
authors use .
and some other
•
In our explanations, we assumed that
these scales are interchangeable without
loss of generality.
•
Have in mind that ReT is computed using
the integral scale l0 and the TKE (related
to the velocity fluctuations around the
mean velocity).
59
Practical turbulence estimates
Can the Kolmogorov eddies become
smaller that the viscous sublayer
length?
60
Practical turbulence estimates
Can the Kolmogorov eddies become smaller that the viscous sublayer length?
•
In a few words, no.
•
In this case, the viscous sublayer is always at least one order of magnitude thinner than the
Kolmogorov eddies.
•
The viscous sublayer cannot accommodate Kolmogorov eddies.
61
Practical turbulence estimates
Can the Kolmogorov eddies become smaller that the viscous sublayer length?
•
As dissipation takes place at the viscous sublayer, it cannot accommodate Kolmogorov eddies.
•
The viscous sublayer will always adapt so it is thinner than the Kolmogorov eddies.
62
Practical turbulence estimates
Can the Kolmogorov eddies become smaller that the viscous sublayer length?
•
As dissipation takes place at the viscous sublayer, it cannot accommodate Kolmogorov eddies.
•
The viscous sublayer will always adapt so it is thinner than the Kolmogorov eddies.
63
Practical turbulence estimates
Can the Kolmogorov eddies become smaller that the viscous sublayer length?
•
As dissipation takes place at the viscous sublayer, it cannot accommodate Kolmogorov eddies.
•
The viscous sublayer will always adapt so it is thinner than the Kolmogorov eddies.
64
Practical turbulence estimates
Can the Kolmogorov eddies become smaller that the viscous sublayer length?
•
To give you an idea how time consuming the postprocessing of large-scale simulations is:
•
We are looking at one timestep of a DNS simulation. The input file is about 17 GB, and it required about 110 GB of RAM memory, a GPU of 16 GB,
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16 cores, and about 5 minutes to open and manipulate the data (mesh size approximately 1.5 billion grid points).
Practical turbulence estimates
Can the Kolmogorov eddies become smaller that the viscous sublayer length?
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Turbulence is a continuum phenomenon.
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The Kolmogorov scales are well above the molecular scales.
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A parameter used to determine when the continuum hypothesis is not valid anymore is the
Knudsen number.
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The Knudsen number is the ratio of the mean free path* to a characteristic dimension (e.g., body
length, turbulent scales, and so on).
Mean free path
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Although there is no definitive criterion, the continuum flow model starts to break down when the
Knudsen number is roughly of the order of 0.1.
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At this Knudsen number values, the flow is rarefied and can not be treated as a continuum.
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In the case of Kn > 0.1, different equations need to be used.
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In these lectures, we will deal only with the continuum equations.
* The mean free path is the average distance travelled by a moving particle (such as an atom, a molecule, a photon) between successive impacts (collisions),
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which modifies its direction or energy or other particle properties.
Practical turbulence estimates
Can the Kolmogorov eddies become smaller that the viscous sublayer length?
•
Although there is no definitive criterion, the continuum flow model starts to break down when the
Knudsen number is roughly of the order of 0.1.
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For Knudsen numbers up to approximately 0.03 the continuum hypothesis is valid.
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For Knudsen numbers between 0.03 and 0.1, the Navier-Stokes equations are valid in the
freestream, but we need to use slip conditions at the walls.
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For Knudsen numbers larger than 0.1 the Navier-Stokes equations cannot be used anymore.
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Practical turbulence estimates
Can the Kolmogorov eddies become smaller that the viscous sublayer length?
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Taking air under atmospheric conditions as an example, the mean free path is 6.5 x 10-8 m (65 nanometers)
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The mean time between successive collision of a molecule is 10-10 s.
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In comparison, the smallest geometric length scale in a flow is seldom less than 0.1 mm (or 10-4 m).
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Which for flow velocities up to 100 m-s-1, yields a flow timescale larger than 10-6 s.
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Thus, even for this example of a flow with small length and time scales, these flow scales exceed the
molecular scales by three or more orders of magnitude.
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This separation can be seen in the plot below, where for very large Reynolds number, the Kolmogorov scales
are well above the molecular scales.
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